Understanding Options

CHAPTER 20

Understanding Options

Answers to Practice Questions

1.a.The put places a floor on value of investment, i.e., less risky than buying stock. The risk reduction comes at the cost of the option premium.

1. Benefit from upside, but also lose on the downside.

1. A naked option position is riskier than the underlying asset. Investor gains from increase in stock price, but loses entire investment if stock price is less than exercise price at expiration.

1. Investor exchanges uncertain upside changes in stock price for the known up-front income from the option premium.

1. Safe investment if the debt is risk free.

1. From put-call parity, this is equivalent (for European options) to ‘buy bond.’ Therefore, this is a safe investment.

1. Another naked, high-risk position with known up-front income but exposure to down movements in stock price.

1. While it is true that both the buyer of a call and the seller of a put hope the price will rise, the two positions are not identical. The buyer of a call will find her profit changing from zero and increasing as the stock price rises [see text Figure 20.1(a)], while the seller of a put will find his loss decreasing and then remaining at zero as the stock price rises [see text Figure 20.2(b)].

1. You would buy the American call for $75, exercise the call immediately in order to purchase a share of Pintail stock for $50, and then sell the share of Pintail stock for $200. The net gain is: $200 – ($75 + $50) = $75.

If the call is a European call, you should buy the call, deposit in the bank an amount equal to the present value of the exercise price, and sell the stock short. This produces a current cash flow equal to: $200 – $75 – ($50/1 + r)

At the maturity of the call, the action depends on whether the stock price is greater than or less than the exercise price. If the stock price is greater than $50, then you would exercise the call (using the cash from the bank deposit) and buy back the stock. If the stock price is less than $50, then you would let the call expire and buy back the stock. The cash flow at maturity is the greater of zero (if the stock price is greater than $50) or [$50 – stock price] (if the stock price is less than $50). Therefore, the cash flows are positive now and zero or positive one year from now.

1. Let P3 = the value of the three month put, C3 = the value of the three month call, S = the market value of a share of stock, and EX = the exercise price of the options. Then, from put-call parity:

C3 + [EX/(1 + r)0.25] = P3 + S

Since both options have an exercise price of $60 and both are worth $10, then:

EX/(1 + r)0.25 = S

From put-call parity for the six-month options, we have:

C6 + [EX/(1 + r)0.50] = P6 + S

Since S = EX/(1 + r)0.25 and EX/(1 + r)0.50 is less than EX/(1 + r)0.25, then the value of the six-month call is greater than the value of the six-month put.

1. From put-call parity:

C + [EX/(1 + r)(2/3)] = P + S

P = –S + C + [EX/(1 + r)(2/3)] = –54 + 9.05 + [50/(1.015(2/3))] = $4.56

1. Internet exercise; answers will vary.

1. The $100 million threshold can be viewed as an exercise price. Since she gains 20% of all profits in excess of this level, it is comparable to a call option. Whether this provides an adequate incentive depends on how achievable the $100 million threshold is and how Ms. Cable evaluates her prospects of generating income greater than this amount.

1. a. The payoffs at expiration for the two options are shown in the following position diagram:

Taking into account the $100 that must be repaid at expiration, the net payoffs are:

1. Here we can use the put-call parity relationship:

Value of call + Present value of exercise price = Value of put + Share price

The value of Mr. Colleoni’s position is:

Value of put (EX = 150) – Value of put (EX = 50) – PV (150 – 50)

Using the put-call parity relationship, we find that this is equal to:

Value of call (EX = 150) – Value of call (EX = 50)

Thus, one combination that gives Mr. Colleoni the same payoffs is:

• Buy a call with an exercise price of $150
• Sell a call with an exercise price of $50

Similarly, another combination with the same set of payoffs is:

• Buy a put with an exercise price of $150
• Buy a share of stock
• Borrow the present value of $150
• Sell a call with an exercise price of $50

1. Statement (b) is correct. The appropriate diagrams are in Figure 20.6 in the text. The first row of diagrams in Figure 20.6 shows the payoffs for the strategy:

Buy a share of stock and buy a put.

The second row of Figure 20.6 shows the payoffs for the strategy:

Buy a call and lend an amount equal to the exercise price.

1. Answers here will vary depending on the options chosen, but the formulas will work very well; discrepancies should be on the order of 5 percent or so, at most.

1. We make use of the put-call parity relationship:

Value of call + Present value of exercise price = Value of put + Share price

1. Rearranging the put-call parity relationship to show a short sale of a share of stock, we have:

(– Share price) = Value of put – Value of call – PV(EX)

This implies that, in order to replicate a short sale of a share of stock, you would purchase a put, sell a call, and borrow the present value of the exercise price.

1. Again we rearrange the put-call parity relationship:

PV(EX) = Value of put - Value of call + Share price

This implies that, in order to replicate the payoffs of a bond, you buy a put, sell a call, and buy the stock.

12.a.Use the put-call parity relationship for European options:

Value of call + Present value of exercise price = Value of put + Share price

Solve for the value of the put:

Value of put = Value of call + PV(EX) – Share price

Thus, to replicate the payoffs for the put, you would buy a 26-week call with an exercise price of $100, invest the present value of the exercise price in a 26-week risk-free security, and sell the stock short.

1. Using the put-call parity relationship, the European put will sell for:

$8 + ($100/1.05) – $90 = $13.24

13.a.From the put-call parity relationship:

Value of call + Present value of exercise price = Value of put + Share price

Equity + PV(Debt, at risk-free rate) = Default option + Assets

$250 + $350 = $70 + $530

b.Value of default put = $350 - $280 = $70

14.Straddle

Butterfly

The buyer of the straddle profits if the stock price moves substantially in either direction; hence, the straddle is a bet on high variability. The buyer of the butterfly profits if the stock price doesn’t move very much, and hence, this is a bet on low variability.

1. Answers here will vary according to the stock and the specific options selected, but all should exhibit properties very close to those predicted by the theory described in the chapter.

1. Imagine two stocks, each with a market price of $100. For each stock, you have an at-the-money call option with an exercise price of $100. Stock A’s price now falls to $50 and Stock B’s rises to $150. The value of your portfolio of call options is now:

Now compare this with the value of an at-the-money call to buy a portfolio with equal holdings of A and B. Since the average change in the prices of the two stocks is zero, the call expires worthless.

This is an example of a general rule: An option on a portfolio is less valuable than a portfolio of options on the individual stocks because, in the latter case, you can choose which options to exercise.

1. Consider each company in turn, making use of the put-call parity relationship:

Value of call + Present value of exercise price = Value of put + Share price

Drongo Corp. Here, the left-hand side [$52 + ($50/1.05) = $99.62] is less than the right-hand side [$20 + $80 = $100]. Therefore, there is a slight mispricing. To take advantage of this situation, one should buy the call, invest $47.62 at the risk-free rate, sell the put, and sell the stock short.

Ragwort, Inc. Here, the left-hand side [$15 + ($100/1.05) = $110.24) is greater than the right-hand side [$10 + $80 = $90]. Therefore, there is a significant mispricing. To take advantage of this situation, one should sell the call, borrow $95.24 at the risk-free rate, buy the put, and buy the stock.

Wombat Corp. For the three-month option, the left-hand side [$18 + ($40/1.025) = $57.02] and the right-hand side [$7 + $50 = $57] are essentially equal, so there is no mispricing.

For the first six-month option, the left-hand side [$17 + ($40/1.05) = $55.10] is slightly greater than the right-hand side [$5 + $50 = $55], so there is a slight mispricing.

For the second six-month option, the left-hand side [$10 + ($50/1.05) = $57.62] is slightly less than the right-hand side [$8 + $50 = $58], and so there is a slight mispricing.

1. One strategy might be to buy a straddle, that is, buy a call and a put with exercise price equal to the asset’s current price. If the asset price does not change, both options become worthless. However, if the price falls, the put will be valuable and, if price rises, the call will be valuable. The larger the price movement in either direction, the greater the profit.

If investors have underestimated volatility, the option prices will be too low. Thus, an alternative strategy is to buy a call (or a put) and hedge against changes in the asset price by simultaneously selling (or, in the case of the put, buying) delta shares of stock.

Challenge Questions

1. a.Purchase a call with a given exercise price and sell a call with a higher exercise price; borrow the difference necessary. (This is known as a ‘Bull Spread.’)

1. Sell a put and sell a call with the same exercise price. (This is known as a ‘Short Straddle.’)

1. Borrow money and use this money to buy a put and buy the stock.

1. Buy one call with a given exercise price, sell two calls with a higher exercise price, and buy one call with a still higher exercise price. (This is known as a ‘Butterfly Spread.’)

2.a.If the land is worth more than $110 million, Bond will exercise its call option. If the land is worth less than $110 million, the buyer will exercise its put option.

b.Bond has: (1) sold a share; (2) sold a put; and (3) purchased a call. Therefore:

This is equivalent to:

1. The interest rate can be deduced using the put-call parity relationship. We know that the call is worth $20, the exercise price is $110, and the combination [sell share and sell put option] is worth $110. Therefore:

Value of call + Present value of exercise price = Value of put + Share price

Value of call + PV(EX) = Value of put + Share price

20 + [110/(1 + r)] = 110

r = 0.222 = 22.2%

1. From the answer to Part (a), we know that Bond will end up owning the land after the expiration of the options. Thus, in an economic sense, the land has not really been sold, and it seems misleading to declare a profit on a sale that did not really take place. In effect, Bond has borrowed money, not sold an asset.

1. One way to profit from Hogswill options is to purchase the call options with exercise prices of $90 and $110, respectively, and sell two call options with an exercise price of $100. The immediate benefit is a cash inflow of:

(2  $11) – ($5 + $15) = $2

Immediately prior to maturity, the value of this position and the net profit (at various possible stock prices) is:

Thus, no matter what the final stock price, we can make a profit trading in these Hogswill options.

It is possible, but very unlikely, that you can identify such opportunities from data published in the newspaper. Someone else has most likely already noticed (even before the paper was printed, much less distributed to you) and traded on the information; such trading tends to eliminate these profit opportunities.

1. a.

1. This incentive scheme is a combination of the following options:

• Buy 4,000,000 call options with an exercise price of $119.875.
• Sell 4,000,000 call options with an exercise price of $120.

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